Thirty-four people were on board the vessel when it collided with another large tourist boat in and capsized on Wednesday at around 10 p.m., CNN reported.
The boats collided in the centre of the city, near the riverside Parliament Building and the Elisabeth Bridge.
Of the 33 South Koreans on board, seven have been rescued, the Ministry said in a statement.
Thirty of the tourists were traveling with South Korean tour company Very Good Tour, the company's chief operating officer Lee Sang-moo said at a press conference.
Lee said the group included nine families, mostly people in their 40s and 50s, although at least one child was on board.
Very Good Tour is planning to send a team of staff to Hungary on Thursday.
Mihaly Toth, a spokesman for boating company Panorama Deck which operated the capsized ship, said the firm is looking into the causes of the capsize.
The tourist boat, called the Mermaid, had been operating on the Danube since 2003 and there had been "no signs of a possible issue with the boat", he added.
South Korean President Moon Jae-in has ordered a rapid response team to be dispatched to help the rescue effort in Budapest, a presidential spokesperson in Seoul said on Thursday.
( With inputs from IANS )